Today, after answering at least a dozen emails from new users about basic design decisions in Sage, I read chapters II and V of Gross and Zagier's seminal paper "Heegner Points and Derivatives of L-series" (at the Vivace coffee shop by REI in downtown Seattle). Chapter II is a very technical derivation of a formula for local archimedian heights of pairings of certain Heegner points. It mainly involves a careful construction of a certain "resolvent kernel function" G by a limiting process, deriving relations between that function and the local archimedian contributions to the height, and dealing with various cases. The function G looks perhaps computable, though I've never heard of anybody computing it; there are some related examples in the last chapter of this paper.
I only skimmed chapters III and IV, which I'm saving for later, since they're 50 pages of very technical arguments.
Chapter V is extremely exciting, since it ties together the ideas from the previous chapters to finally relate heights of Heegner points to L-values. The basic idea is that using Rankin's method, Gross and Zagier write down a "horrendous" formula for the coefficients of a certain modular form that encodes the linear functional that sends a newform f to the value of the derivative at 1 of a certain L-series. They also use that G function mentioned above and explicit computation with local heights to write down another horrendous formulas for height pairings of Heegner points. They then observe in Chapter V that the formulas as the same! (And, yes, they include an excited exclamation point at that point in the paper.) The rest of Chapter V involves deducing the statements that originally motivated Gross's interest in this problem (e.g., the conjectures of Birch related to the BSD conjecture), an application to weight 1, and some ruminations on higher weight analogues that involve Grothendieck motives attached to higher weight modular forms, though Gross-Zagier talk only of Deligne's cohomology theory, since this paper was written before Scholl's paper on motives attached to modular forms. Much of these later deductions will be very fun to generalize to higher derivatives, since they're basically formal applications of their main formula.
Idea/Question: Something I've long wondered about is whether there is a way to compute the *conjectural* order of Sha and the regulator for modular abelian varieties A over the rational numbers with positive rank. Reading Chapter V of Gross-Zagier makes me think there is in the common case when rank(A) = dim(A), which would be incredibly useful for making a large table that generalizes Cremona's book to dimension bigger than 1. Here's the very roughly sketched idea. Given a newform f of degree > 1 over the rational numbers, there is a corresponding abelian variety A=A_f of dimension d > 1. It is fairly easy to compute L(A,s), and even the leading coefficient L^(d)(A,1) and BSD volume Omega_A. Assuming A has rank d, according to the BSD conjecture, L^(d)(A,1)/Omega_A is a nonzero rational multiple alpha of the regulator. I wonder if one can get any information at all -- even conjecturally -- about this multiple by computing something explicit involving the various constructions in the Gross-Zagier paper? Some sort of calculation using Gross-Zagier should give the regulator of the subgroup generated by the Heegner point and its Galois conjugates, and this will be a square multiple of the true regulator. Comparing that with what BSD gives for Reg*Sha, and throwing in knowledge of (or bounds on) the Tamagawa numbers and torsion, and an analogue of the conjecture in Section 5 that the index of the Heegner subgroup in the full group of rational points is a formula involving the square root of something involving the torsion, Tamagawa numbers, Manin constant, and order of Sha, should yield a conjectural formula for Sha. This would take working out stuff not given explicitly in Gross-Zagier in higher dimension, but I think would be do-able. This will definitely require explicitly computing something about p-divisibility or indexes of Heegner points in A(K), possibly using ideas from Johan Bosman's Ph.D. thesis.
I talked more with Robert Bradshaw about all the above, and our conclusion was a first interesting problem is just to figure out precisely what replaces the elliptic curve index [E(K):Z y_K] when E is replaced by A=A_f. Maybe [A_f(K) : T y_K] where T is the Hecke algebra? But then sqrt(#Sha(E))*prod c_p * Torsion(E) has to be replaced by something else since #Sha(A) might not be a perfect square.